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 Markov random fields, Gibbs states and entropy minimality
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Markov random fields, Gibbs states and entropy minimality Chandgotia, Nishant
Abstract
The wellknown HammersleyClifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbour interaction. Following Petersen and Schmidt we utilise the formalism of cocycles for the homoclinic relation and introduce "Markov cocycles", reparametrisations of Markov specifications. We exploit this formalism to deduce the conclusion of the HammersleyClifford Theorem for a family of Markov random fields which are outside the theorem's purview (including Markov random fields whose support is the ddimensional "3colored chessboard"). On the other extreme, we construct a family of shiftinvariant Markov random fields which are not given by any finite range shiftinvariant interaction. The techniques that we use for this problem are further expanded upon to obtain the following results: Given a "fourcycle free" finite undirected graph H without selfloops, consider the corresponding 'vertex' shift, H ơm(Zd,H) denoted by X(H). We prove that X(H) has the pivot property, meaning that for all distinct configurations x,y ∈ X(H) which differ only at finitely many sites there is a sequence of configurations (x=x¹),x²,...,(xn =y) ∈ X(H) for which the successive configurations (xi,xi+1) differ exactly at a single site. Further if H is connected we prove that X(H) is entropy minimal, meaning that every shift space strictly contained in X(H) has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the 'lifts' of the configurations in X(H) to their universal cover and the introduction of 'height functions' in this context. Further we generalise the HammersleyClifford theorem with an added condition that the underlying graph is bipartite. Taking inspiration from Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong configfolding to prove that if all Markov random fields supported on X are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the "strong configfolds" and "strong configunfolds" of X.
Item Metadata
Title 
Markov random fields, Gibbs states and entropy minimality

Creator  
Publisher 
University of British Columbia

Date Issued 
2015

Description 
The wellknown HammersleyClifford Theorem states (under certain conditions) that any Markov random field is a Gibbs state for a nearest neighbour interaction. Following Petersen and Schmidt we utilise the formalism of cocycles for the homoclinic relation and introduce "Markov cocycles", reparametrisations of Markov specifications. We exploit this formalism to deduce the conclusion of the HammersleyClifford Theorem for a family of Markov random fields which are outside the theorem's purview (including Markov random fields whose support is the ddimensional "3colored chessboard"). On the other extreme, we construct a family of shiftinvariant Markov random fields which are not given by any finite range shiftinvariant interaction.
The techniques that we use for this problem are further expanded upon to obtain the following results: Given a "fourcycle free" finite undirected graph H without selfloops, consider the corresponding 'vertex' shift, H ơm(Zd,H) denoted by X(H). We prove that X(H) has the pivot property, meaning that for all distinct configurations x,y ∈ X(H) which differ only at finitely many sites there is a sequence of configurations (x=x¹),x²,...,(xn =y) ∈ X(H) for which the successive configurations (xi,xi+1) differ exactly at a single site. Further if H is connected we prove that X(H) is entropy minimal, meaning that every shift space strictly contained in X(H) has strictly smaller entropy.
The proofs of these seemingly disparate statements are related by the use of the 'lifts' of the configurations in X(H) to their universal cover and the introduction of 'height functions' in this context.
Further we generalise the HammersleyClifford theorem with an added condition that the underlying graph is bipartite. Taking inspiration from Brightwell and Winkler we introduce a notion of folding for configuration spaces called strong configfolding to prove that if all Markov random fields supported on X are Gibbs with some nearest neighbour interaction so are Markov random fields supported on the "strong configfolds" and "strong configunfolds" of X.

Genre  
Type  
Language 
eng

Date Available 
20150421

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivs 2.5 Canada

DOI 
10.14288/1.0166259

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201505

Campus  
Scholarly Level 
Graduate

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Aggregated Source Repository 
DSpace

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Rights
AttributionNonCommercialNoDerivs 2.5 Canada